Zero-Sum $K_m$ Over ${{{\mathbb {Z}}}}$ and the Story of $K_4$

Abstract

We prove the following results solving a problem raised by Caro and Yuster (Graphs Comb 32:49–63, 2016). For a positive integer $m\ge 2$, $m\ne 4$, there are infinitely many values of n such that the following holds: There is a weighting function $f:E(K_n)\rightarrow \{-1,1\}$ (and hence a weighting function $f: E(K_n)\rightarrow \{-1,0,1\}$), such that $\sum _{e\in E(K_n)}f(e)=0$ but, for every copy H of $K_m$ in $K_n$, $\sum _{e\in E(H)}f(e)\ne 0$. On the other hand, for every integer $n\ge 5$ and every weighting function $f:E(K_n)\rightarrow \{-1,1\}$ such that $|\sum _{e\in E(K_n)}f(e)|\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) - 2h(n)$, where $h(n)=(n+1)$ if $n \equiv 0$ (mod 4) and $h(n)=n$ if $n \not \equiv 0$ (mod 4), there is always a copy H of $K_4$ in $K_n$ for which $\sum _{e\in E(H)}f(e)=0$, and the value of h(n) is sharp.